Hamiltonicity of Sparse Pseudorandom Graphs

We show that every $(n,d,\lambda)$-graph contains a Hamilton cycle for sufficiently large $n$, assuming that $d\geq \log^{10}n$ and $\lambda\leq cd$, where $c=\frac{1}{9000}$. This significantly improves a recent result of Glock, Correia and Sudakov, who obtain a similar result for $d$ that grows polynomially with $n$. The proof is based on the absorption technique combined with a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices. We believe that the latter result is of an independent interest and will have further applications.